Sparse principal component analysis (PCA) aims at mapping large dimensional
data to a linear subspace of lower dimension. By imposing loading vectors to be
sparse, it performs the double duty of dimension reduction and variable
selection. Sparse PCA algorithms are usually expressed as a trade-off between
explained variance and sparsity of the loading vectors (i.e., number of
selected variables). As a high explained variance is not necessarily synonymous
with relevant information, these methods are prone to select irrelevant
variables. To overcome this issue, we propose an alternative formulation of
sparse PCA driven by the false discovery rate (FDR). We then leverage the
Terminating-Random Experiments (T-Rex) selector to automatically determine an
FDR-controlled support of the loading vectors. A major advantage of the
resulting T-Rex PCA is that no sparsity parameter tuning is required. Numerical
experiments and a stock market data example demonstrate a significant
performance improvement.

稀疏主成分分析通过将高维数据映射到低维线性子空间，通过设置加载向量为稀疏，实现降维和变量选择；然而，为了避免选择无关变量，本研究提出了使用假阳性发现率控制负载向量支撑的稀疏主成分分析算法，并结合 “终止随机实验” 选择器来自动确定支撑集合；实验结果表明这种方法显著提高了性能。

具有伪发现率控制变量选择的稀疏主成分分析

Sparse PCA with False Discovery Rate Controlled Variable Selection

The autoencoder is an effective unsupervised learning model which is widely
used in deep learning. It is well known that an autoencoder with a single
fully-connected hidden layer, a linear activation function and a squared error
cost function trains weights that span the same subspace as the one spanned by
the principal component loading vectors, but that they are not identical to the
loading vectors. In this paper, we show how to recover the loading vectors from
the autoencoder weights.

本文介绍了使用一种单隐层全连接自编码器进行特征提取可以有效地恢复主成分分析 (PCA) 的加载向量，且训练权重与 PCA 加载向量存在差异，从而优化特征提取的性能。

利用线性自编码器从主子空间提取主成分

From Principal Subspaces to Principal Components with Linear  Autoencoders

Principal component analysis (PCA) is a widely used technique for data
analysis and dimension reduction with numerous applications in science and
engineering. However, the standard PCA suffers from the fact that the principal
components (PCs) are usually linear combinations of all the original variables,
and it is thus often difficult to interpret the PCs. To alleviate this
drawback, various sparse PCA approaches were proposed in literature [15, 6, 17,
28, 8, 25, 18, 7, 16]. Despite success in achieving sparsity, some important
properties enjoyed by the standard PCA are lost in these methods such as
uncorrelation of PCs and orthogonality of loading vectors. Also, the total
explained variance that they attempt to maximize can be too optimistic. In this
paper we propose a new formulation for sparse PCA, aiming at finding sparse and
nearly uncorrelated PCs with orthogonal loading vectors while explaining as
much of the total variance as possible. We also develop a novel augmented
Lagrangian method for solving a class of nonsmooth constrained optimization
problems, which is well suited for our formulation of sparse PCA. We show that
it converges to a feasible point, and moreover under some regularity
assumptions, it converges to a stationary point. Additionally, we propose two
nonmonotone gradient methods for solving the augmented Lagrangian subproblems,
and establish their global and local convergence. Finally, we compare our
sparse PCA approach with several existing methods on synthetic, random, and
real data, respectively. The computational results demonstrate that the sparse
PCs produced by our approach substantially outperform those by other methods in
terms of total explained variance, correlation of PCs, and orthogonality of
loading vectors.

本研究提出了一种新的稀疏 PCA 方法，旨在找到稀疏和几乎不相关的主成分，并具有正交的载荷向量，同时尽可能多地解释总方差。我们还开发了一种新的增广 Lagrangian 方法来解决一类非光滑约束优化问题，该方法非常适合我们的稀疏 PCA 公式。最后，我们将我们的稀疏 PCA 方法与其他方法在合成数据，随机数据和真实数据上进行比较。计算结果表明，我们的方法产生的稀疏主成分在总方差，主成分相关性和载荷向量的正交性等方面显着优于其他方法。